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<h1><a
 href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><IMG
 align=middle alt=[delaunay] height=100
 src="qh--dt.gif" width=100 ></a> Qhull format options (F)</h1>

<p>This section lists the format options for Qhull. These options
are indicated by 'F' followed by a letter. See <A
 href="qh-opto.htm#output" >Output</a>, <A href="qh-optp.htm#print">Print</a>,
and <A href="qh-optg.htm#geomview">Geomview</a> for other output
options. </p>

<p><b>Copyright &copy; 1995-2012 C.B. Barber</b></p>

<hr>

<p><A href="index.htm#TOC">&#187;</a> <A href="qh-quick.htm#programs">Programs</a>
<a name="format">&#149;</a> <A href="qh-quick.htm#options">Options</a>
&#149; <A href="qh-opto.htm#output">Output</a>
&#149; <A href="qh-optf.htm#format">Formats</a>
&#149; <A href="qh-optg.htm#geomview">Geomview</a>
&#149; <A href="qh-optp.htm#print">Print</a>
&#149; <A href="qh-optq.htm#qhull">Qhull</a>
&#149; <A href="qh-optc.htm#prec">Precision</a>
&#149; <A href="qh-optt.htm#trace">Trace</a></p>

<h2>Additional input &amp; output formats</h2>

<p>These options allow for automatic processing of Qhull output.
Options '<A href="qh-opto.htm#i">i</a>', '<A href="qh-opto.htm#o">o</a>',
'<A href="qh-opto.htm#n">n</a>', and '<A href="qh-opto.htm#p">p</a>'
may also be used.</p>

<dl compact>
    <dt>
    <dd><b>Summary and control</b>
    <dt><A href="#FA">FA</a>
    <dd>compute total area and volume for option '<A
        href="qh-opto.htm#s">s</a>'

    <dt><A href="#FV">FV</a>
    <dd>print average vertex (interior point for '<A
        href="qhalf.htm">qhalf</a>')
    <dt><A href="#FQ">FQ</a>
    <dd>print command for qhull and input
    <dt><A href="#FO">FO</a>
    <dd>print options to stderr or stdout
    <dt><A href="#FS">FS</a>
    <dd>print sizes: total area and volume
    <dt><A href="#Fs">Fs</a>
    <dd>print summary: dim, #points, total vertices and
  facets, #vertices, #facets, max outer and inner plane
    <dt><A href="#Fd">Fd</a>
    <dd>use  format for input (offset first)
    <dt><A href="#FD">FD</a>
    <dd>use cdd format for normals (offset first)
    <dt><a href="#FM">FM</a>
    <dd>print Maple output (2-d and 3-d)
    <dt>
    <dt>
    <dd><b>Facets, points, and vertices</b>
    <dt><A href="#Fa">Fa</a>
    <dd>print area for each facet
    <dt><A href="#FC">FC</a>
    <dd>print centrum for each facet
    <dt><A href="#Fc">Fc</a>
    <dd>print coplanar points for each facet
    <dt><A href="#Fx">Fx</a>
    <dd>print extreme points (i.e., vertices) of convex hull.

    <dt><A href="#FF">FF</a>
    <dd>print facets w/o ridges
    <dt><A href="#FI">FI</a>
    <dd>print ID for each facet
    <dt><A href="#Fi">Fi</a>
    <dd>print inner planes for each facet
    <dt><A href="#Fm">Fm</a>
    <dd>print merge count for each facet (511 max)
    <dt><A href="#FP">FP</a>
    <dd>print nearest vertex for coplanar points
    <dt><A href="#Fn">Fn</a>
    <dd>print neighboring facets for each facet
    <dt><A href="#FN">FN</a>
    <dd>print neighboring facets for each point
    <dt><A href="#Fo">Fo</a>
    <dd>print outer planes for each facet
    <dt><A href="#Ft">Ft</a>
    <dd>print triangulation with added points
    <dt><A href="#Fv">Fv</a>
    <dd>print vertices for each facet
    <dt>
    <dt>
    <dd><b>Delaunay, Voronoi, and halfspace</b>
    <dt><A href="#Fx">Fx</a>
    <dd>print extreme input sites of Delaunay triangulation
  or Voronoi diagram.
    <dt><A href="#Fp">Fp</a>
    <dd>print points at halfspace intersections
    <dt><A href="#Fi2">Fi</a>
    <dd>print separating hyperplanes for inner, bounded
  Voronoi regions
    <dt><A href="#Fo2">Fo</a>
    <dd>print separating hyperplanes for outer, unbounded
  Voronoi regions
    <dt><A href="#Fv2">Fv</a>
    <dd>print Voronoi diagram as ridges for each input pair
    <dt><A href="#FC">FC</a>
    <dd>print Voronoi vertex ("center") for each facet</dd>
</dl>

<hr>

<h3><A href="#format">&#187;</a><a name="Fa">Fa - print area for each
facet </a></h3>

<p>The first line is the number of facets. The remaining lines
are the area for each facet, one facet per line. See '<A
 href="#FA" >FA</a>' and '<A href="#FS">FS</a>' for computing the total area and volume.</p>

<p>Use '<A href="qh-optp.htm#PAn">PAn</a>' for printing the n
largest facets. Use option '<A href="qh-optp.htm#PFn">PFn</a>'
for printing facets larger than <i>n</i>.</p>

<p>For Delaunay triangulations, the area is the area of each
Delaunay triangle. For Voronoi vertices, the area is the area of
the dual facet to each vertex. </p>

<p>Qhull uses the centrum and ridges to triangulate
non-simplicial facets. The area for non-simplicial facets is the
sum of the areas for each triangle. It is an approximation of the
actual area. The ridge's vertices are projected to the facet's
hyperplane. If a vertex is far below a facet (qh_WIDEcoplanar in <tt>user.h</tt>),
the corresponding triangles are ignored.</p>

<p>For non-simplicial facets, vertices are often below the
facet's hyperplane. If so, the approximation is less than the
actual value and it may be significantly less. </p>

<h3><A href="#format">&#187;</a><a name="FA">FA - compute total area
and volume for option 's' </a></h3>

<p>With option 'FA', Qhull includes the total area and volume in
the summary ('<A href="qh-opto.htm#s">s</a>').  Option '<A href="#FS">FS</a>' also includes the total area and volume.
If facets are
merged, the area and volume are approximations. Option 'FA' is
automatically set for options '<A href="#Fa">Fa</a>', '<A
 href="qh-optp.htm#PAn" >PAn</a>', and '<A href="qh-optp.htm#PFn">PFn</a>'.
</p>

<p>With '<A href="qdelaun.htm">qdelaunay</a> <A
 href="qh-opto.htm#s" >s</a> FA', Qhull computes the total area of
the Delaunay triangulation. This equals the volume of the convex
hull of the data points. With options '<A href="qdelau_f.htm">qdelaunay Qu</a>
<A href="qh-opto.htm#s">s</a> FA', Qhull computes the
total area of the furthest-site Delaunay triangulation. This
equals of the total area of the Delaunay triangulation. </p>

<p>See '<A href="#Fa">Fa</a>' for further details.  Option '<A href="#FS">FS</a>' also computes the total area and volume.</p>

<h3><A href="#format">&#187;</a><a name="Fc">Fc - print coplanar
points for each facet </a></h3>

<p>The output starts with the number of facets. Then each facet
is printed one per line. Each line is the number of coplanar
points followed by the point ids. </p>

<p>By default, option 'Fc' reports coplanar points
('<A href="qh-optq.htm#Qc">Qc</a>').  You may also use
option '<A href="qh-optq.htm#Qi">Qi</a>'. Options 'Qi Fc' prints
interior points while 'Qci Fc' prints both coplanar and interior
points.

<p>Each coplanar point or interior point is assigned to the
facet it is furthest above (resp., least below). </p>

<p>Use 'Qc <A href="qh-opto.htm#p">p</a>' to print vertex and
coplanar point coordinates. Use '<A href="qh-optf.htm#Fv">Fv</a>'
to print vertices. </p>

<h3><A href="#format">&#187;</a><a name="FC">FC - print centrum or
Voronoi vertex for each facet </a></h3>

<p>The output starts with the dimension followed by the number of
facets. Then each facet centrum is printed, one per line. For
<A href="qvoronoi.htm">qvoronoi</a>, Voronoi vertices are
printed instead. </p>

<h3><A href="#format">&#187;</a><a name="Fd">Fd - use cdd format for
input </a></h3>

<p>The input starts with comments. The first comment is reported
in the summary. Data starts after a "begin" line. The
next line is the number of points followed by the dimension plus
one and "real" or "integer". Then the points
are listed with a leading "1" or "1.0". The
data ends with an "end" line.</p>

<p>For halfspaces ('<A href="qhalf.htm">qhalf</a> Fd'),
the input format is the same. Each halfspace starts with its
offset. The signs of the offset and coefficients are the
opposite of Qhull's
convention. The first two lines of the input may be an interior
point in '<A href="#FV">FV</a>' format.</p>

<h3><A href="#format">&#187;</a><a name="FD">FD - use cdd format for
normals </a></h3>

<p>Option 'FD' prints normals ('<A href="qh-opto.htm#n">n</a>', '<A
 href="#Fo" >Fo</a>', '<A href="#Fi">Fi</a>') or points ('<A
 href="qh-opto.htm#p" >p</a>') in cdd format. The first line is the
command line that invoked Qhull. Data starts with a
"begin" line. The next line is the number of normals or
points followed by the dimension plus one and "real".
Then the normals or points are listed with the offset before the
coefficients. The offset for points is 1.0. For normals,
the offset and coefficients use the opposite sign from Qhull.
The data ends with an "end" line.</p>

<h3><A href="#format">&#187;</a><a name="FF">FF - print facets w/o
ridges </a></h3>

<p>Option 'FF' prints all fields of all facets (as in '<A
 href="qh-opto.htm#f" >f</a>') without printing the ridges. This is
useful in higher dimensions where a facet may have many ridges.
For simplicial facets, options 'FF' and '<A href="qh-opto.htm#f">f
</a>' are equivalent.</p>

<h3><A href="#format">&#187;</a><a name="Fi">Fi - print inner planes
for each facet </a></h3>

<p>The first line is the dimension plus one. The second line is
the number of facets. The remainder is one inner plane per line.
The format is the same as option '<A href="qh-opto.htm#n">n</a>'.</p>

<p>The inner plane is a plane that is below the facet's vertices.
It is an offset from the facet's hyperplane. It includes a
roundoff error for computing the vertex distance.</p>

<p>Note that the inner planes for Geomview output ('<A
 href="qh-optg.htm#Gi" >Gi</a>') include an additional offset for
vertex visualization and roundoff error. </p>

<h3><A href="#format">&#187;</a><a name="Fi2">Fi - print separating
hyperplanes for inner, bounded Voronoi regions</a></h3>

<p>With <A href="qvoronoi.htm" >qvoronoi</a>, 'Fi' prints the
separating hyperplanes for inner, bounded regions of the Voronoi
diagram. The first line is the number of ridges. Then each
hyperplane is printed, one per line. A line starts with the
number of indices and floats. The first pair of indices indicates
an adjacent pair of input sites. The next <i>d</i> floats are the
normalized coefficients for the hyperplane, and the last float is
the offset. The hyperplane is oriented toward '<A
 href="qh-optq.htm#QVn" >QVn</a>' (if defined), or the first input
site of the pair. </p>

<p>Use '<A href="qh-optf.htm#Fo2">Fo</a>' for unbounded regions,
and '<A href="qh-optf.htm#Fv2">Fv</a>' for the corresponding
Voronoi vertices. </p>

<p>Use '<A href="qh-optt.htm#Tv">Tv</a>' to verify that the
hyperplanes are perpendicular bisectors. It will list relevant
statistics to stderr. The hyperplane is a perpendicular bisector
if the midpoint of the input sites lies on the plane, all Voronoi
vertices in the ridge lie on the plane, and the angle between the
input sites and the plane is ninety degrees. This is true if all
statistics are zero. Roundoff and computation errors make these
non-zero. The deviations appear to be largest when the
corresponding Delaunay triangles are large and thin; for example,
the Voronoi diagram of nearly cospherical points. </p>

<h3><A href="#format">&#187;</a><a name="FI">FI - print ID for each
facet </a></h3>

<p>Print facet identifiers. These are used internally and listed
with options '<A href="qh-opto.htm#f">f</a>' and '<A href="#FF">FF</a>'.
Options '<A href="#Fn">Fn </a>' and '<A href="#FN">FN</a>' use
facet identifiers for negative indices. </p>

<h3><A href="#format">&#187;</a><a name="Fm">Fm - print merge count
for each facet </a></h3>

<p>The first line is the number of facets. The remainder is the
number of merges for each facet, one per line. At most 511 merges
are reported for a facet. See '<A href="qh-optp.htm#PMn">PMn</a>'
for printing the facets with the most merges. </p>

<h3><a href="#format">&#187;</a><a name="FM">FM - print Maple
output </a></h3>

<p>Qhull writes a Maple file for 2-d and 3-d convex hulls,
2-d and 3-d halfspace intersections,
and 2-d Delaunay triangulations. Qhull produces a 2-d
or 3-d plot.

<p><i>Warning</i>: This option has not been tested in Maple.

<p>[From T. K. Abraham with help from M. R. Feinberg and N. Platinova.]
The following steps apply while working within the
Maple worksheet environment :
<ol>
<li>Generate the data and store it as an array . For example, in 3-d, data generated
in Maple is of the form : x[i],y[i],z[i]
<p>
<li>Create a single variable and assign the entire array of data points to this variable.
Use the "seq" command within square brackets as shown in the following example.
(The square brackets are essential for the rest of the steps to work.)
<p>
>data:=[seq([x[i],y[i],z[i]],i=1..n)]:# here n is the number of data points

<li>Next we need to write the data to a file to be read by qhull. Before
writing the data to a file, make sure that the qhull executable files and
the data file lie in   the same subdirectory. If the executable files are
stored in the "C:\qhull3.1\" subdirectory, then save the file in the same
subdirectory, say "C:\qhull3.1\datafile.txt". For the sake of integrity of
the data file , it is best to first ensure that the data file does not
exist before writing into the data file. This can be done by running a
delete command first . To write the data to the file, use the "writedata"
and the "writedata[APPEND]" commands as illustrated in the following example :
<p>
>system("del c:\\qhull3.1\\datafile.txt");#To erase any previous versions of the file
<br>>writedata("c:\\qhull3.1\\datafile.txt ",[3, nops(data)]);#writing in qhull format
<br>>writedata[APPEND]("c:\\ qhull3.1\\datafile.txt ",   data);#writing the data points
<li>
Use the 'FM' option to produce Maple output.  Store the output as a ".mpl" file.
For example, using the file we created above, we type the following (in DOS environment)
<p>
qconvex s FM &lt;datafile.txt >dataplot.mpl

<li>
To read 3-d output in Maple, we use the 'read'  command followed by
a 'display3d' command.  For example (in Maple environment):
<p>
>with (plots):
<br>>read `c:\\qhull3.1\\dataplot.mpl`:#IMPORTANT - Note that the punctuation mark used is ' and NOT '. The correct punctuation   mark is the one next to the key for "1" (not the punctuation mark near the enter key)
<br>> qhullplot:=%:
<br>> display3d(qhullplot);
</ol>

<p>For Delaunay triangulation orthogonal projection is better.

<p>For halfspace intersections, Qhull produces the dual
convex hull.

<p>See <a href="qh-faq.htm#math">Is Qhull available for Maple?</a>
for other URLs.

<h3><A href="#format">&#187;</a><a name="Fn">Fn - print neighboring
facets for each facet </a></h3>

<p>The output starts with the number of facets. Then each facet
is printed one per line. Each line is the number of neighbors
followed by an index for each neighbor. The indices match the
other facet output formats.</p>

<p>For simplicial facets, each neighbor is opposite
the corresponding vertex (option '<A href="#Fv">Fv</a>').
Do not compare to option '<A href="qh-opto.htm#i">i</a>'.  Option 'i'
orients facets by reversing the order of two vertices.  For non-simplicial facets,
the neighbors are unordered.

<p>A negative index indicates an unprinted facet due to printing
only good facets ('<A href="qh-optp.htm#Pg">Pg</a>', <A href="qdelaun.htm" >qdelaunay</a>,
<A href="qvoronoi.htm" >qvoronoi</a>). It
is the negation of the facet's ID (option '<A href="#FI">FI</a>').
For example, negative indices are used for facets "at
infinity" in the Delaunay triangulation.</p>

<h3><A href="#format">&#187;</a><a name="FN">FN - print neighboring
facets for each point </a></h3>

<p>The first line is the number of points. Then each point is
printed, one per line. For unassigned points (either interior or
coplanar), the line is "0". For assigned coplanar
points ('<A href="qh-optq.htm#Qc">Qc</a>'), the line is
"1" followed by the index of the facet that is furthest
below the point. For assigned interior points ('<A
 href="qh-optq.htm#Qi" >Qi</a>'), the line is "1"
followed by the index of the facet that is least above the point.
For vertices that do not belong to good facet, the line is
"0"</p>

<p>For vertices of good facets, the line is the number of
neighboring facets followed by the facet indices. The indices
correspond to the other '<A href="#format">F</a>' formats. In 4-d
and higher, the facets are sorted by index. In 3-d, the facets
are in adjacency order (not oriented).</p>

<p>A negative index indicates an unprinted facet due to printing
only good facets (<A href="qdelaun.htm" >qdelaunay</a>,
<A href="qvoronoi.htm" >qvoronoi</a>, '<A href="qh-optp.htm#Pdk">Pdk</a>',
'<A href="qh-optp.htm#Pg">Pg</a>'). It is the negation of the
facet's ID ('<A href="#FI"> FI</a>'). For example, negative
indices are used for facets "at infinity" in the
Delaunay triangulation.</p>

<p>For Voronoi vertices, option 'FN' lists the vertices of the
Voronoi region for each input site. Option 'FN' lists the regions
in site ID order. Option 'FN' corresponds to the second half of
option '<A href="qh-opto.htm#o">o</a>'. To convert from 'FN' to '<A
 href="qh-opto.htm#o" >o</a>', replace negative indices with zero
and increment non-negative indices by one. </p>

<p>If you are using the <A href="qh-code.htm#library">Qhull
library</a> or <A href="qh-code.htm#cpp">C++ interface</a>, option 'FN' has the side effect of reordering the
neighbors for a vertex </p>

<h3><A href="#format">&#187;</a><a name="Fo">Fo - print outer planes
for each facet </a></h3>

<p>The first line is the dimension plus one. The second line is
the number of facets. The remainder is one outer plane per line.
The format is the same as option '<A href="qh-opto.htm#n">n</a>'.</p>

<p>The outer plane is a plane that is above all points. It is an
offset from the facet's hyperplane. It includes a roundoff error
for computing the point distance. When testing the outer plane
(e.g., '<A href="qh-optt.htm#Tv">Tv</a>'), another roundoff error
should be added for the tested point.</p>

<p>If outer planes are not checked ('<A href="qh-optq.htm#Q5">Q5</a>')
or not computed (!qh_MAXoutside), the maximum, computed outside
distance is used instead. This can be much larger than the actual
outer planes.</p>

<p>Note that the outer planes for Geomview output ('<A
 href="qh-optg.htm#G" >G</a>') include an additional offset for
vertex/point visualization, 'lines closer,' and roundoff error.</p>

<h3><A href="#format">&#187;</a><a name="Fo2">Fo - print separating
hyperplanes for outer, unbounded Voronoi regions</a></h3>

<p>With <A href="qvoronoi.htm" >qvoronoi</a>, 'Fo' prints the
separating hyperplanes for outer, unbounded regions of the
Voronoi diagram. The first line is the number of ridges. Then
each hyperplane is printed, one per line. A line starts with the
number of indices and floats. The first pair of indices indicates
an adjacent pair of input sites. The next <i>d</i> floats are the
normalized coefficients for the hyperplane, and the last float is
the offset. The hyperplane is oriented toward '<A
 href="qh-optq.htm#QVn" >QVn</a>' (if defined), or the first input
site of the pair. </p>

<p>Option 'Fo' gives the hyperplanes for the unbounded rays of
the unbounded regions of the Voronoi diagram. Each hyperplane
goes through the midpoint of the corresponding input sites. The
rays are directed away from the input sites. </p>

<p>Use '<A href="qh-optf.htm#Fi2">Fi</a>' for bounded regions,
and '<A href="qh-optf.htm#Fv2">Fv</a>' for the corresponding
Voronoi vertices. Use '<A href="qh-optt.htm#Tv">Tv</a>' to verify
that the corresponding Voronoi vertices lie on the hyperplane. </p>

<h3><A href="#format">&#187;</a><a name="FO">FO - print list of
selected options </a></h3>

<p>Lists selected options and default values to stderr.
Additional 'FO's are printed to stdout. </p>

<h3><A href="#format">&#187;</a><a name="Fp">Fp - print points at
halfspace intersections</a></h3>

<p>The first line is the number of intersection points. The
remainder is one intersection point per line. A intersection
point is the intersection of <i>d</i> or more halfspaces from
'<A href="qhalf.htm">qhalf</a>'. It corresponds to a
facet of the dual polytope. The "infinity" point
[-10.101,-10.101,...] indicates an unbounded intersection.</p>

<p>If [x,y,z] are the dual facet's normal coefficients and <i>b&lt;0</i>
is its offset, the halfspace intersection occurs at
[x/-b,y/-b,z/-b] plus the interior point. If <i>b&gt;=0</i>, the
halfspace intersection is unbounded. </p>

<h3><A href="#format">&#187;</a><a name="FP">FP - print nearest
vertex for coplanar points </a></h3>

<p>The output starts with the number of coplanar points. Then
each coplanar point is printed one per line. Each line is the
point ID of the closest vertex, the point ID of the coplanar
point, the corresponding facet ID, and the distance. Sort the
lines to list the coplanar points nearest to each vertex. </p>

<p>Use options '<A href="qh-optq.htm#Qc">Qc</a>' and/or '<A
 href="qh-optq.htm#Qi" >Qi</a>' with 'FP'. Options 'Qc FP' prints
coplanar points while 'Qci FP' prints coplanar and interior
points.  Option 'Qc' is automatically selected if 'Qi' is not
selected.

<p>For Delaunay triangulations (<A href="qdelaun.htm" >qdelaunay</a>
or <A href="qvoronoi.htm" >qvoronoi</a>), a coplanar point is nearly
incident to a vertex. The distance is the distance in the
original point set.</p>

<p>If imprecision problems are severe, Qhull will delete input
sites when constructing the Delaunay triangulation.  Option 'FP' will
list these points along with coincident points.</p>

<p>If there are many coplanar or coincident points and non-simplicial
facets are triangulated ('<A href="qh-optq.htm#Qt">Qt</a>'), option
'FP' may be inefficient.  It redetermines the original vertex set
for each coplanar point.</p>

<h3><A href="#format">&#187;</a><a name="FQ">FQ - print command for
qhull and input </a></h3>

<p>Prints qhull and input command, e.g., "rbox 10 s | qhull
FQ".   Option 'FQ' may be repeated multiple times.</p>

<h3><A href="#format">&#187;</a><a name="Fs">Fs - print summary</a></h3>

<p>The first line consists of number of integers ("10")
followed by the:
<ul>
<li>dimension
<li>number of points
<li>number of vertices
<li>number of facets
<li>number of vertices selected for output
<li>number of facets selected for output
<li>number of coplanar points for selected facets
<li>number of nonsimplicial or merged facets selected for
  output
<LI>number of deleted vertices</LI>
<LI>number of triangulated facets ('<A href="qh-optq.htm#Qt">Qt</a>')</LI>
</ul>

<p>The second line consists of the number of reals
("2") followed by the:
<ul>
<li>maximum offset to an outer plane
<li>minimum offset to an inner plane.</li>
</ul>
Roundoff and joggle are included.
<P></P>

<p>For Delaunay triangulations and Voronoi diagrams, the
number of deleted vertices should be zero. If greater than zero, then the
input is highly degenerate and coplanar points are not necessarily coincident
points.  For example, <tt>'RBOX 1000 s W1e-13 t995138628 | QHULL d Qbb'</tt> reports
deleted vertices; the input is nearly cospherical.</p>

<P>Later versions of Qhull may produce additional integers or reals.</P>

<h3><A href="#format">&#187;</a><a name="FS">FS - print sizes</a></h3>

<p>The first line consists of the number of integers
("0"). The second line consists of the number of reals
("2"), followed by the total facet area, and the total
volume. Later versions of Qhull may produce additional integers
or reals.</p>

<p>The total volume measures the volume of the intersection of
the halfspaces defined by each facet. It is computed from the
facet area. Both area and volume are approximations for
non-simplicial facets. See option '<A href="#Fa">Fa </a>' for
further notes. Option '<A href="#FA">FA </a>' also computes the total area and volume. </p>

<h3><A href="#format">&#187;</a><a name="Ft">Ft - print triangulation</a></h3>

<p>Prints a triangulation with added points for non-simplicial
facets. The output is </p>

<ul>
    <li>The first line is the dimension
    <li>The second line is the number of points, the number
  of facets, and the number of ridges.
    <li>All of the input points follow, one per line.
    <li>The centrums follow, one per non-simplicial facet
    <li>Then the facets follow as a list of point indices
        preceded by the number of points. The simplices are
        oriented. </li>
</ul>

<p>For convex hulls with simplicial facets, the output is the
same as option '<A href="qh-opto.htm#o">o</a>'.</p>

<p>The added points are the centrums of the non-simplicial
facets. Except for large facets, the centrum is the average
vertex coordinate projected to the facet's hyperplane. Large
facets may use an old centrum to avoid recomputing the centrum
after each merge. In either case, the centrum is clearly below
neighboring facets. See <A href="qh-impre.htm">Precision issues</a>.
</p>

<p>The new simplices will not be clearly convex with their
neighbors and they will not satisfy the Delaunay property. They
may even have a flipped orientation. Use triangulated input ('<A
 href="qh-optq.htm#Qt">Qt</a>') for Delaunay triangulations.

<p>For Delaunay triangulations with simplicial facets, the output is the
same as option '<A href="qh-opto.htm#o">o</a>' without the lifted
coordinate.  Since 'Ft' is invalid for merged Delaunay facets, option
'Ft' is not available for qdelaunay or qvoronoi.  It may be used with
joggled input ('<A  href="qh-optq.htm#QJn" >QJ</a>') or triangulated output ('<A
 href="qh-optq.htm#Qt" >Qt</a>'), for example, rbox 10 c G 0.01 | qhull d QJ Ft</p>

<p>If you add a point-at-infinity with '<A href="qh-optq.htm#Qz">Qz</a>',
it is printed after the input sites and before any centrums. It
will not be used in a Delaunay facet.</p>

<h3><A href="#format">&#187;</a><a name="Fv">Fv - print vertices for
each facet</a></h3>

<p>The first line is the number of facets. Then each facet is
printed, one per line. Each line is the number of vertices
followed by the corresponding point ids. Vertices are listed in
the order they were added to the hull (the last one added is the
first listed).
</p>
<p>Option '<A href="qh-opto.htm#i">i</a>' also lists the vertices,
but it orients facets by reversing the order of two
vertices.  Option 'i' triangulates non-simplicial, 4-d and higher facets by
adding vertices for the centrums.
</p>

<h3><A href="#format">&#187;</a><a name="Fv2">Fv - print Voronoi
diagram</a></h3>

<p>With <A href="qvoronoi.htm" >qvoronoi</a>, 'Fv' prints the
Voronoi diagram or furthest-site Voronoi diagram. The first line
is the number of ridges. Then each ridge is printed, one per
line. The first number is the count of indices. The second pair
of indices indicates a pair of input sites. The remaining indices
list the corresponding ridge of Voronoi vertices. Vertex 0 is the
vertex-at-infinity. It indicates an unbounded ray. </p>

<p>All vertices of a ridge are coplanar. If the ridge is
unbounded, add the midpoint of the pair of input sites. The
unbounded ray is directed from the Voronoi vertices to infinity. </p>

<p>Use '<A href="qh-optf.htm#Fo2">Fo</a>' for separating
hyperplanes of outer, unbounded regions. Use '<A
 href="qh-optf.htm#Fi2" >Fi</a>' for separating hyperplanes of
inner, bounded regions. </p>

<p>Option 'Fv' does not list ridges that require more than one
midpoint. For example, the Voronoi diagram of cospherical points
lists zero ridges (e.g., 'rbox 10 s | qvoronoi Fv Qz').
Other examples are the Voronoi diagrams of a rectangular mesh
(e.g., 'rbox 27 M1,0 | qvoronoi Fv') or a point set with
a rectangular corner (e.g.,
'rbox P4,4,4 P4,2,4 P2,4,4 P4,4,2 10 | qvoronoi Fv').
Both cases miss unbounded rays at the corners.
To determine these ridges, surround the points with a
large cube (e.g., 'rbox 10 s c G2.0 | qvoronoi Fv Qz').
The cube needs to be large enough to bound all Voronoi regions of the original point set.
Please report any other cases that are missed. If you
can formally describe these cases or
write code to handle them, please send email to <A
 href="mailto:qhull@qhull.org" >qhull@qhull.org</a>. </p>

<h3><A href="#format">&#187;</a><a name="FV">FV - print average
vertex </a></h3>

<p>The average vertex is the average of all vertex coordinates.
It is an interior point for halfspace intersection. The first
line is the dimension and "1"; the second line is the
coordinates. For example,</p>

<blockquote>
    <p>qconvex FV <A
    href="qh-opto.htm#n">n</a> | qhalf <A href="#Fp">Fp</a></p>
</blockquote>

<p>prints the extreme points of the original point set (roundoff
included). </p>

<h3><A href="#format">&#187;</a><a name="Fx">Fx - print extreme
points (vertices) of convex hulls and Delaunay triangulations</a></h3>

<p>The first line is the number of points. The following lines
give the index of the corresponding points. The first point is
'0'. </p>

<p>In 2-d, the extreme points (vertices) are listed in
counterclockwise order (by qh_ORIENTclock in user.h). </p>

<p>In 3-d and higher convex hulls, the extreme points (vertices)
are sorted by index. This is the same order as option '<A
 href="qh-opto.htm#p" >p</a>' when it doesn't include coplanar or
interior points. </p>

<p>For Delaunay triangulations, 'Fx' lists the extreme
points of the input sites (i.e., the vertices of their convex hull). The points
are unordered. <!-- Navigation links --> </p>

<hr>

<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull<br>
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<hr>

<p><a href="http://www.geom.uiuc.edu/"><IMG align=middle
 height=40 src="qh--geom.gif" width=40 ></a><i>The Geometry Center
Home Page </i></p>

<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
</a><br>
Created:
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